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Abstract

In x-ray diffraction microscopy, iterative algorithms retrieve reciprocal space phase information, and a real space image, from an object’s coherent diffraction intensities through the use of a priori information such as a finite support constraint. In many experiments, the object’s shape or support is not well known, and the diffraction pattern is incompletely measured. We describe here computer simulations to look at the effects of both of these possible errors when using several common reconstruction algorithms. Overly tight object supports prevent successful convergence; however, we show that this can often be recognized through pathological behavior of the phase retrieval transfer function. Dynamic range limitations often make it difficult to record the central speckles of the diffraction pattern. We show that this leads to increasing artifacts in the image when the number of missing central speckles exceeds about 10, and that the removal of unconstrained modes from the reconstructed image is helpful only when the number of missing central speckles is less than about 50. This simulation study helps in judging the reconstructability of experimentally recorded coherent diffraction patterns.

Illustration of the various support errors studied. A blue area denotes a region added to the correct support. A red area denotes a region removed from the correct support. (a) Magnitude of a reconstructed cell image from diffraction pattern with 2×2 pixel missing center. (b) The correct support. (c) The “bump-out” support generated by including 1439 pixels (out of 400×400) outside the correct support at a local area. (d) The “bite-in” support generated by excluding 1382 pixels inside the correct support at a local area. (e) The “loose” support generated by increasing the size of the correct support uniformly by 2 pixels which adds 1477 pixels. (f) The “tight” support generated by reducing the correct support size uniformly by 2 pixels which removes 1439 pixels.

Magnitudes of reconstructed images of the σ = 4.6 simulated cell from different algorithms with varying sizes of missing centers. Only the central part of diffraction patterns and reconstructed images are displayed. The green line indicates where artifacts becomes noticeable in reconstructed images.

Image magnitudes before and after unconstrained modes were removed from HIO reconstructions with various missing center sizes for σ = 4.6 cell (top row) and σ = 7.5 cell (bottom row). Up to 13 missing speckles, the artifacts in reconstructed images are negligible in both cases. Unconstrained mode removal works well up to 21 missing speckles for the σ = 4.6 cell, and 30 for the σ = 7.5 cell. Above that level, artifacts cannot be completely removed. When the missing speckle number is greater than 48, permanent artifacts are present in images in both cases.

Measurements of the effects of missing central speckles in the measured diffraction intensities. Shown here are both the signal-to-noise ratio with unconstrained modes removed (red curve), and Rreal calculated before (grey curve) and after (black curve) removal of unconstrained missing modes. The results are shown for the same size simulated cell in a smaller array (σ = 4.6) at left, and a larger array (σ = 7.5) at right.

Metrics

Table 1

Real space r factor Rreal and signal-to-noise ratio SNR of reconstructed images with different supports. Smaller Rreal and larger SNR indicate more reliable reconstructions. The best image quality is highlighted for each case